Omstein-Uhlenbeck Processes

In die literature on stochastic processes, the above Fokker-Planck equation describes a multi-variate Omstein— Uhlenbeck process. For a discussion on the existence of Gaussian solutions to this process, see Gardiner (1990). 

E. The Fractional Omstein-Uhlenbeck Process IV. The Fractional Kramers Escape Problem... 

The first moment of the fractional Omstein-Uhlenbeck process can be calculated from Eq. (58). It evolves in time like... 

Figure 5. Mean squared displacement for the fractional (a = 1 /2, full line) and normal (dashed) Omstein-Uhlenbeck process. The Brownian process shows the typical proportionality to t for small times it approaches the saturation value much faster than its subdiffusive analogue, which starts off with the r /2 behavior and approaches the thermal equilibrium value by a power law, compare Eq. (61)...

Finally we would like to mention the work of Lavenda and coworkers, who approached the same problem by means of a generalization of the kinetic analog of Boltzmann s principle so as to include the Omstein-Uhlenbeck process. They carried out an asymptotic analysis in the limit of high resistance. The condition for the validity of their asymptotic expansion has been proved to be identical to the modified Kramers condition derived by Stratonovich. 

In the case a = 1, the Mittag-Leffler function becomes the exponential, so that the solution to the fractional Langevin equation reduces to that for an Omstein-Uhlenbeck process... 

Diffusion. - Distribution of the diffusivitity of fluid in a horizontally oriented cylinder was demonstrated by NMR imaging in two papers on a granular flow system and in the earth s magnetic field. Correlation time (ic) and diffusion coefficient (D = Xc) imaging (CTDCI) was applied to a granular flow system of 2 mm oil-filled sphere rotated in a half-filled horizontal cylinder, ie. to an Omstein-Uhlenbeck process with a velocity autocorrelation function. Time dependent apparent diffusion coefficients are measured, and Tc... 

Schobel R, Zhu JW (1999) Stochastic Volatility with an Omstein-Uhlenbeck Process an Extension. European Finance Review 3 23-46. 

In the Vasicek (1977) model, the instantaneous short-rate r is assumed to follow a stochastic process known as the Omstein-Uhlenbeck process, a form of Gaussian process, described by Equation (3.24) ... 

Equations [25] describe free diffusion of the center-of-mass and N independent Omstein-Uhlenbeck processes. The formal solution of these eqirations is... 

In the diffusion limit it is foimd that the combined effects of particle inertia and shear flow modify the amphtude and the time-dependence of the particle-velocity autocorrelation functions, a result which is expressed in terms of the Stokes number, St = 7/fi. The shear flow breaks macroscopic time reversibility and stationarity the autocorrelation functions of the particle velocities are stationary and the velocity correlation along the shear is symmetric in the time difference t, but the cross correlation is non-symmetric in t function in the streamwise direction is non-stationary The time decay of the velocity correlation along the flow is not a pure exponential and the imderlying stochastic process is not an Omstein-Uhlenbeck process. 

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